Re: Cantor Diagonal Proof

wij <wyniijj5@gmail.com> writes:

On Thu, 2025-04-10 at 17:23 -0700, Keith Thompson wrote:

wij <wyniijj5@gmail.com> writes:
[...]

"lim(x->c) f(x)=L" means the limit of f approaching c is L, not f(c)=L 'eventually'.
f at c is not defined (handled) in limit.

 
Correct.
 

lim 0.333...=1/3    ... The *limit* is 1/3, not 0.333...=1/3
0.3+0.33+0.333+...  ... The sequence converges to 1/3
Σ(n=1,∞) 3/10^n     ... The sum converges to 1/3 (or you can use lim)

 
The limit as the number of 3s increases without bound *is exactly what
we mean* by the notation "0.333...".  Once you understand that, it's
obvious that 0.333... is exactly equal to 1/3, and that 0.333... is a
rational number.

>
You agree "f at c is not defined (handled) in limit", yet, on the other hand
ASSERTING 0.333... is 'exactly' 1/3 from limit? Are you nut?
 
As usual, you need to prove what you say. Or you are just showing yourself
another olcott, just blink belief, nothing else.

Keep the insults to yourself.  Last warning.

My assertion is simply about what the "..." notation means.

Do you agree that the limit of 0.3, 0.33, 0.333, as the number of 3s
increases without bound, is exactly 1/3?  (You said so above.)

What exactly do you think the notation "0.333..." means?  I and many
others use that notation to mean the limit, which you agree is exactly
1/3.  Why do you object to the use of that particular notation for that
particular concept?

Do you have any mathematical argument that isn't purely about notation
that you dislike?

Would, say "0.333<∞>" be clearer?  Could you agree that that refers to
the limit and gives a result that's exactly 1/3?  If so, why do you
object to "..." but not to "<∞>" as a symbol for the limit?  (Note that
"..." is easier to type, unless you happen to have an ∞ key your
keyboard.)

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
void Void(void) { Void(); } /* The recursive call of the void */